exponentially
exponentially — 副詞
1. at an increasingly rapid rate where the amount multiplies instead of adding a fi
指數級
以極快速度不斷倍增的方式
at an increasingly rapid rate where the amount multiplies instead of adding a fixed quantity, so growth speeds up dramatically over time
Omar saw his company's monthly sales grow exponentially in the first two years.
Omar 的公司月銷售額在頭兩年呈指數級成長。
collocation: grow exponentially
The number of Jisoo's app users increased exponentially after it won a design award.
Jisoo 的語言學習應用程式獲獎後，使用人數呈指數級成長。
collocation: increase exponentially
Owen's after-school origami club grew exponentially as members taught their friends how to fold paper cranes.
Owen 的課後摺紙社團成員每週教新朋友摺紙鶴，社團人數呈指數級成長。
Anong realized that computing power had risen exponentially while costs dropped at the same time.
Anong 發現運算能力呈指數級提升，而成本同時下降。
- rapidly
describes speed but does not imply multiplication; more general and less intense
- dramatically
emphasizes the surprising or impressive nature of the change but not the mathematical pattern
- sharply
suggests a steep, sudden change rather than gradual multiplication over time
- gradually
implies slow, step-by-step change rather than accelerating multiplication
- marginally
describes a very small change, the opposite of the dramatic growth pattern
文法句型
exponentially + increase/grow/rise/expand
用法筆記
Frequently paired with verbs of increase such as grow, rise, expand, multiply, and improve. Not used with stative verbs (know, like, own). The emphasis is always on repeated multiplication, not steady addition.
常見錯誤
exponentially — 形容詞
- exponentiallypositive
- more exponentiallycomparative
- most exponentiallysuperlative
1. relating to the small raised number that tells you how many times to multiply a
指數的
與指數數學符號有關的
relating to the small raised number that tells you how many times to multiply a base number by itself — for example, in 2³ the 3 is an exponential symbol showing 2 × 2 × 2
Kwame wrote exponential notation on the board to show that 5⁴ means 5 × 5 × 5 × 5.
Kwame 在黑板上寫下指數記法，說明 5⁴ 就是 5×5×5×5。
example: exponential notation for repeated multiplication
Kemi wrote 7³ in her lab report to avoid repeating 7 × 7 × 7 on every line.
Kemi 在實驗報告中寫下 7³，省去每行重複寫 7×7×7 的麻煩。
Madison used exponential symbols to shorten the repeated multiplication in her homework.
Madison 用指數符號來縮寫作業中重複的乘法算式。
Ilan's calculator showed huge numbers in exponential format to save screen space.
Ilan 的計算機用指數格式顯示很大的數字，以節省畫面空間。
用法筆記
Domain: mathematics. This sense refers specifically to the written notation of exponents (the small raised number). It is distinct from sense 3, which describes the function or growth pattern.
常見錯誤
2. describing a mathematical expression in which the exponent is a variable — such
指數變數
以變數做為指數的
describing a mathematical expression in which the exponent is a variable — such as 2ˣ or aⁿ — rather than a fixed number
Nikhil studied exponential equations where the unknown letter appeared in the exponent position.
Nikhil 研究了未知數出現在指數位置的指數方程式。
subject + verb + exponential + noun: exponential equations
Kian saw that 3ˣ was an exponential expression with the variable in the exponent.
Kian 認出 3ˣ 這個式子是一種指數形式，變數寫在底數的上方。
Eleni solved for x in the exponential equation 5ˣ = 125 by rewriting the base on both sides.
Eleni 將兩邊底數化為相同，解出指數方程式 5ˣ = 125 中的 x。
Ravindra stared at 2ⁿ = 64 on his algebra quiz and tried to find the value of n.
Ravindra 盯著代數小考題目 2ⁿ = 64，試圖找出 n 的值。
用法筆記
Domain: algebra. Unlike sense 1, which focuses on the notation, this sense is about expressions where the exponent itself is a variable. These are often taught alongside logarithmic functions.
3. relating to a mathematical function in which the rate of change is proportional
指數函數
可用指數函數表達的
relating to a mathematical function in which the rate of change is proportional to the current value — representable by a formula such as f(x) = a·bˣ — producing multiplicative growth or decay over equal time periods
Vinícius drew an exponential curve showing bacteria doubling every hour.
Vinícius 在圖表上畫出一條指數曲線，顯示細菌數量每小時倍增。
collocation: exponential curve + concrete scenario (bacteria)
Eve modelled the spread of the virus using an exponential function with a base of 1.3.
Eve 用底數為 1.3 的指數函數來模擬病毒的傳播。
Astrid watched the ant colony double every month and drew an exponential growth curve for her project.
Astrid 觀察蟻群每月數量倍增，並畫出指數成長曲線完成專題報告。
Emily used an exponential function to calculate loan growth over ten years.
Emily 用指數函數計算貸款在十年間的成長。
用法筆記
Domain: mathematics and sciences. This sense is the closest to the everyday meaning of exponential growth. Distinguish from sense 2: sense 2 focuses on expressions with a variable exponent, while this sense focuses on the functional relationship and its graph.